The amplitude of e^ e^ - i is equal to - Vidyadip

MCQ

The amplitude of ${e^{{e^{ - i\theta }}}}$is equal to

A
$\sin \theta $
✓
$ - \sin \theta $
C
${e^{\cos \theta }}$
D
${e^{\sin \theta }}$

✓

Answer

Correct option: B.

$ - \sin \theta $

b
(b)Let $z = {e^{{e^{ - i\theta }}}} = {e^{\cos \theta - i\sin \theta }}$$ = {e^{\cos \theta }}{e^{ - i\sin \theta }}$
$z = {e^{\cos \theta }}[\cos (\sin \theta ) - i\sin (\sin \theta )]$
$z = {e^{\cos \theta }}\cos (\sin \theta ) - i{e^{\cos \theta }}\sin (\sin \theta )$
$amp(z) = {\tan ^{ - 1}}\left[ { - \frac{{{e^{\cos \theta }}\sin (\sin \theta )}}{{{e^{\cos \theta }}\cos (\sin \theta )}}} \right]$
$ = {\tan ^{ - 1}}[\tan ( - \sin \theta )] = - \sin \theta $.

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